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P erturbative Results Without Diagrams R. Rosenfelder Paul-Scherrer-Institut, Villigen PSI (Switzerland) PSI, 31 July 2008 arXiv:0805.4525 [hep-th] , submitted to Phys. Rev. E (Computational Physics) 1. Introduction 2. A new method (applied to

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SLIDE 1

Perturbative Results Without Diagrams

  • R. Rosenfelder

Paul-Scherrer-Institut, Villigen PSI (Switzerland)

PSI, 31 July 2008

arXiv:0805.4525 [hep-th] , submitted to Phys. Rev. E (Computational Physics)

  • 1. Introduction
  • 2. A new method (applied to the polaron g.s. energy)
  • 3. Numerical procedures and results
  • 4. Summary
  • 5. Outlook: application to worldline QED
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  • R. Rosenfelder (PSI) :

PT without diagrams 2

  • 1. Introduction

Usually in perturbative calculation in Quantum (Field) Theory the number

  • f diagrams grows factorially with the order

Example: number of diagrams for g-2 of the electron in QED

(see Itzykson & Zuber p. 466, 467) Γ(α) = 4z(1 − S) S3 , S = −2z

  • 1 + K′

0(z)

K0(z)

  • , z = − 1

4α expand in powers of α = 1/137.036 = ⇒ Γ(α) = 1 + α + 7 α2 + 72 α3 + 891 α4 + 12672 α5 + 202770 α6 + . . .

Consequence: huge cancellations between individual diagrams heroic efforts needed for higher-order calculations

Schwinger (1948), Petermann, Sommerfield (1957) Laporta & Remiddi (1996), Kinoshita et al. (1990-2005)

Need (more modestly: would be nice to have) new methods !

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  • R. Rosenfelder (PSI) :

PT without diagrams 3

  • 2. A new method (applied to the polaron g. s. energy)

Take as simple (but nontrivial) example the polaron problem – a non-relativistic field theory

polaron = electron slowly moving through polarizable crystal model Hamiltonian

  • H. Fr¨
  • hlich (1954)

ˆ H ∼ 1 2ˆ

p2 +

  • k

ˆ a†

kˆ

ak + √α

  • k

1 |k|

  • ˆ

a†

ke−ik·ˆ x + h.c.

  • α: dimensionless electron-phonon coupling constant

Ground-state energy of polaron: E0 : =

  • n=1

en αn

,

e1 = −1 e2

= −0.01591962 (1959) ,

e3 = −0.00080607

Smondyrev (1986)

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  • R. Rosenfelder (PSI) :

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In field-theoretic language: have to evaluate self-energy diagrams with more and more loops Long live the PATH INTEGRAL : phonons can be integrated out exactly! Feynman (1955) Z(β) =

  • D3x e−Seff

β→∞

− → e−βE0 where for large β ⇒ Seff[x] ∼

β

dt 1 2 ˙

x2 + α β

dt1dt2 e−|t1−t2|

  • d3k 1

k2 exp [ik · (x(t1) − x(t2))] = : S0 + S1

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  • R. Rosenfelder (PSI) :

PT without diagrams 5

Employ cumulant expansion of partition function Z(β) = Z0 exp

  • n=1

(−)n n! λn(β)

  • where λn(β) are the cumulants w.r.t. S1

Recursion relation with the moments mn(β) ≡ S1n ∝ αn λn+1 = mn+1 −

n−1

  • k=0

n

k

  • λk+1 mn−k

λ1 = m1 λ2 = m2 − m2

1

λ3 = m3 − 3 m2 m1 + 2 m3

1

λ4 = m4 − 4 m3 m1 − 3 m2

2 + 12 m2 m2 1 − 6 m4 1

λ5 = m5 − 5 m4 m1 − 10 m3 m2 + 20 m3 m2

1 + 30 m2 2 m1 − 60 m2 m3 1 + 24 m5 1

. . .

Note : λn(β) ∝ αn

= ⇒

en = lim

β→∞

1 β (−)n+1 αn n! λn(β)

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  • R. Rosenfelder (PSI) :

PT without diagrams 6

The path integral for the moments mn = C

  • D3x S1n e−S0[x] ,

m0 = 1 can be evaluated exactly. Write Coulomb propagator as 1

k2 = 1

2

∞

du exp

  • −1

2k2 u

  • =

⇒ all momentum integrations can be performed and one obtains mn = (−)n αn (4π)n/2

n

  • m=1

β

dtm

tm

dt′

m

∞

dum

  • exp
  • −

n

  • m=1

(tm − t′

m)

  • ·

det A t1 . . . tn, t′

1 . . . t′ n; u1 . . . un

−3/2

with (n × n)- matrix A

Aij

= 1 2

  • −|ti − tj| + |ti − t′

j| + |t′ i − tj| − |t′ i − t′ j|

  • + ui δij

↑ ↑ non-analytic analytic dependence

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SLIDE 7
  • R. Rosenfelder (PSI) :

PT without diagrams 7 Define

Aij = : aij + ui δij

Diagonal parts:

aii = ti − t′

i ≡ σi

Non-diagonal matrix elements : S := 1 2

  • ti + t′

i − (tj + t′ j)

r := 1 2 (σi − σj) s := 1 2 (σi + σj)

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  • R. Rosenfelder (PSI) :

PT without diagrams 8

  • 3. Numerical procedures and results

For mn(β) ⇒ λn(β) one has to do a 3n-dimensional integral over ti, t′

i, ui

Two ui-integrations can be done analytically

————————————————————————

∞

dun det−3/2

n

A(1, 2, . . . , n) = 2 An

  • detn A(un = 0)

∞

dun−1

∞

dun det−3/2

n

A(1, 2, . . . , n) = 4

  • An−1,n An−1 An

arcsin √xHF √xHF where An, An−1, An−1,n are principal minors of the determinant detn A ≡ A 0 ≤ xHF : = 1 − An−1,n A An−1 An ≤ 1 because Aij is a positive semi-definite matrix = ⇒ Hadamard-Fischer inequality An−1 An ≥ An−1,n A

————————————————————————

After performing un, un−1-integrations analytically = ⇒ (3n − 2)-dimensional integral left

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  • R. Rosenfelder (PSI) :

PT without diagrams 9

Useful trick: calculate directly ∂λn

∂β =

⇒ (3n − 3)-dimensional integral ! Further advantage: asymptotic behaviour (thus extrapolation to β → ∞ ) is much improved:

en(β) : = (−)n+1

αn n! ∂λn ∂β = ∂ ∂β

  • β · en + const

× − . . .

β→∞

− → en − an √β e−β (?) + . . . Exponential convergence to en : analytically proved for n = 1, 2 numerically for n = 3 : assume en(β) → en − an β−κn e−β fit to Monte-Carlo data gives κ3 = 0.55(3) Assume it also for n > 3 ...

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  • R. Rosenfelder (PSI) :

PT without diagrams 10

Numerical evaluation: mapping to hypercube [0, 1] , then: Monte-Carlo integration with VEGAS program or routines from the CUBA library

Note: Monte-Carlo integration can handle non-analytic, even discontinous integrands

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  • R. Rosenfelder (PSI) :

PT without diagrams 11

check n = 3 (6-dimensional integral):

analytical: e3 = −0.80607005·10−3

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  • R. Rosenfelder (PSI) :

PT without diagrams 12

but for n = 4 convergence is slow with number of function calls: solution: perform the (n − 2) remaining ui -integrations by deterministic integration routine. Very efficient: tanh-sinh-method !

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  • R. Rosenfelder (PSI) :

PT without diagrams 13 − → Transformation x = g(t) = tanh ( sinh t) t ∈ [−∞, +∞] g′(t) = cosh t cosh2(sinh t) Euler − MacLaurin = ⇒

+1

−1

dx f(x) =

+∞

−∞

dt g′(t) f (g(t)) ≈ h

k=+∞

  • k=−∞

wk f (xk) with xk = g(kh) , wk = g′(kh) wk double exponentially decreasing for large |k|

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  • R. Rosenfelder (PSI) :

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now for n = 4:

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  • R. Rosenfelder (PSI) :

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and also n = 5 (evaluation of (9+3)-dim. integral) is within reach:

Here tanh-sinh-integration seems to be slightly better than Gauss-Legendre

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  • R. Rosenfelder (PSI) :

PT without diagrams 16

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  • R. Rosenfelder (PSI) :

PT without diagrams 17

  • 4. Summary
  • Two additional perturbative coefficients e4 , e5 for the polaron g.s. energy have been

determined by a new (mostly) numerical method. This amounts to performing a 4-loop and 5-loop calculation in Quantum Field Theory

  • Method is based on a combination of Monte-Carlo integration techniques and deter-

ministic quadrature rules for finite β (temperature) and on judicious extrapolation to β → ∞ (zero temperature). Reproduces Smondyrev’s coefficient e3 with high accuracy

  • Cancellation in nth order not among many individual diagrams but among much fewer

terms in the integrand of the (3n − 3)-dimensional integral

  • Increased computational power would allow to improve accuracy for e4 , e5 and even

e6 seems accessible

  • Application to g-2 of the electron under investigation (worldline representation of

QED). Challenge: renormalization !?

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  • R. Rosenfelder (PSI) :

PT without diagrams 18

  • 5. Outlook: application to worldline QED

Generating functional Z[¯ η, η, j] =

  • D ¯

ψDψDA exp

  • i S[ ¯

ψ, ψ, A] + ( ¯ ψ, η) + (¯ η, ψ) + (j, A)

  • S[ ¯

ψ, ψ, A] =

  • ¯

ψ,

  • γ · (i∂ − eA
  • ≡Π

) − M0

  • ψ
  • + S0[A]

2-point function:

  • ψ(x) ¯

ψ(0) ∼

  • DA
  • x
  • 1

γ · Π − M0

  • eiS0[A] Det (γ · Π − M0)
  • ↓

↑ Schwinger trick = const. in quenched approx. ↓ ∼

∞

dT

  • dχ exp

i (γ · Π)2 − M2

  • T − i (γ · Π + M0) χ
  • dχ = 0 ,
  • dχ χ = 1

Berezin

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  • R. Rosenfelder (PSI) :

PT without diagrams 19

result in momentum space : (Alexandrou, RR & Schreiber, PR A 59 (1999))

G2(p) ∼

∞

dT

  • dχ e−i(M 2

0 T+M0χ)

  • d4x e−ip·x
  • DA eiS0[A]
  • D4x
  • D4ζ ei S[x,ζ,χ,A]
  • Γ→γ
  • rbital trajectory : x(0) = 0 , x(T) = x

spin trajectory : ζ(0) + ζ(T) = Γ S[x, ζ, χ, A] ∼

T

dt

  • −1

2 ˙ x2 + iζ · ˙ ζ + ˙ x · ζ χ − e ˙ x · A − ie ζ · F · ζ

  • ↑

↑ ↑

spin-orbit convection spin current

Photon field A can be integrated out exactly = ⇒ effective action

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  • R. Rosenfelder (PSI) :

PT without diagrams 20

In supersymmetric formulation: t → t, θ Xµ :∼ xµ + θ ζµ , D := ∂ ∂θ − θ ∂ ∂t Seff[x, ζ, χ] ∼ S0 + α

  • dt1dθ1 dt2dθ2
  • d4k 1

k2 DX1 · DX2 exp [−ik · (X1 − X2)] ———————————— տ free photon propagator

2-point function G2(p) ∼

∞

dT

  • dχ exp
  • i A(p, Γ, T) + iB(p, Γ, T) χ
  • develops pole at γ · p = Mphys if for large T

A(p, Γ, T) − →

  • p2 − M2

phys

  • T

B(p, Γ, T) − → Z2 ( p · Γ + Mphys )

————————————– Determine Mphys and wavefunction renorm. Z2 – similar to polaron case ! From 3-point function < ψ ¯ ψA > : electron-photon coupling, i.e. anomalous magnetic moment of electron